chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2023-24-4-78-84cheb-1596Research ArticleСтатьиArticleХроматическое число слоек с запрещенными одноцветными арифметическими прогрессиямиOn the chromatic number of slices without monochromatic unit arithmetic progressionsКироваВалерия ОрлановнаKirovaValeria Orlanovnakirova_vo@mail.ruМосковский государственный университет им. М. В. ЛомоносоваРоссияLomonosov Moscow State UniversityRussian Federation2023250120242447884Copyright © Кирова В.О., 20232023Кирова В.О.Kirova V.O.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/1596

Для ℎ, 𝑛 ≥ 1 и 𝑒 > 0 рассматривается хроматическое число пространств вида R^𝑛×[0, 𝑒]^ℎ.Представлен обзор имеющихся результатов, рассмотрена задача о хроматическом численормированных пространств с запрещенными одноцветными арифметическими прогрес-сиями. Показано, что для любого 𝑛 существует двуцветная раскраска пространства R^𝑛,при которой достаточно длинная арифметическая прогрессия содержит точки обоих цветов, и такая раскраска применима к пространствам вида R^𝑛 × [0, 𝑒]^ℎ.

For ℎ, 𝑛 ≥ 1 and 𝑒 > 0 we consider a chromatic number of the spaces R^𝑛×[0, 𝑒]^ℎ and general results in this problem. Also we consider the chromatic number of normed spaces with forbidden monochromatic arithmetic progressions. We show that for any 𝑛 there exists a two-coloring of R^𝑛 such that all long unit arithmetic progressions contain points of both colors and this coloring covers spaces of the form R^𝑛×[0,𝑒]^ℎ.

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The authors declare that there are no conflicts of interest present.